{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R487",
  "content_hash": null,
  "slug": "m127ac-open-three-step-gap",
  "type": "claim",
  "title": "Three shorter lengths remain unresolved",
  "summary": "Settling the target requires either a chain of length at most 135 or a certificate excluding all three shorter lengths.",
  "relevance": "For Shortest addition chain for the 127th Mersenne number, record m127ac-open-three-step-gap (“Three shorter lengths remain unresolved”) records a bound, answer, status fact, or structural consequence. The record states: Settling the target requires either a chain of length at most 135 or a certificate excluding all three shorter lengths.",
  "relevance_source": "recorded",
  "body": "The 136-step construction does not prove optimality. The lower-bound audit excludes lengths through 132 according to the reported exhaustive Hamming-weight result. It leaves lengths 133, 134, and 135.\n\nA complete solution can take either form: an explicit shorter chain, or an exhaustive certificate ruling out each remaining length. Any future lower-bound computation should publish enough reduced-graph, defect-pattern, or search-frontier data for independent replay.",
  "status": "open",
  "evidence_grade": "supported",
  "scope": {
    "kind": "bounded",
    "statement": "exclusion or construction of addition chains of lengths 133, 134, and 135 for N=2^127-1",
    "bounds": {
      "candidate_length": {
        "min": 133,
        "max": 135
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://additionchains.com/",
      "locator": "Gap obtained by combining the current lower-bound audit with the replayed upper-bound certificate"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://additionchains.com/",
    "locator": "Gap obtained by combining the current lower-bound audit with the replayed upper-bound certificate"
  },
  "relations": [
    {
      "slug": "R486",
      "title": "The strongest located interval is 133 through 136",
      "object_type": "claim",
      "relation": "refines",
      "direction": "outgoing"
    },
    {
      "slug": "mersenne-127-addition-chain",
      "title": "mersenne 127 addition chain",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}